1. (a) 5 (b) -7 (c) -15
2. (a) Domain \(x \ge -5\), range \(y \ge 0\) (b) Domain \(x \neq 3\), range \(y \neq 0\) (c) Domain \([-2,2]\), range \([-4,0]\)
3. (a) Yes (b) No (c) Yes
4. (a) 74 (b) 52 (c) \(25x - 6\)
5. (a) \(f^{-1}(x) = \frac{2x - 4}{3}\) (b) \(f^{-1}(x) = \sqrt{x + 1}\)
6. Vertex \((3,0)\), intercepts \((0,6)\), \((6,6)\)
7. Domain \([9, \infty)\)
8. Proof using definition of inverse and composition
9. \(f^{-1}(x) = \frac{ax + b}{cx - a}\) (self-inverse)
10. (a) Yes (linear, \(m \neq 0\)) (b) \(C^{-1}(c) = \frac{c - 200}{1.5}\) (c) 100 minutes
1. (a) \((x - 3)^2 + 1\), min = 1 at \(x = 3\) (b) \(-2(x - 2)^2 + 3\), max = 3 at \(x = 2\) (c) \(3(x + 2)^2 - 5\), min = -5 at \(x = -2\)
2. (a) 49, two real (b) 0, one repeated (c) -23, no real
3. (a) \(x < 1\) or \(x > 3\) (b) \(-4 \le x \le 2\)
4. \(2(x - 2)^2 - 3\), min = -3 at \(x = 2\)
5. (b) Max area 100 m² at \(x = 10\) (square) (c) \(5 \le x \le 15\)
6. \(k < -2\) or \(k > 6\)
7. \(c = -1\)
8. (b) \(k = 1\)
9. (a) \(m^2 + 4m - 12 < 0\) (b) \(-6 < m < 2\)
10. (a) 6000 subscribers (b) Rs 64,000 (c) \(x = 2\) and \(x = 10\) (d) \(2 < x < 10\)
1. (a) \(x = 2, 8\) (b) \(x = -5, 4\) (c) \(x = -\frac{4}{3}, 4\)
2. (a) \(x < -10\) or \(x > 2\) (b) \(-1 \le x \le \frac{7}{3}\)
3. Roots at 2, 4, 6
4. \(x = -1\) or \(x = 4\)
5. \(0.5 \le d \le 3.5\) km
6. \(x < 1\) or \(3 < x < 5\)
7. \(\frac{2}{3} \le x \le 8\)
8. \(k = -2\)
9. \(x = 1, 2, 3, 4\)
10. (a) \(k = -2\) (c) \(x \le 1\) or \(3 \le x \le 5\)
11. (b) \(2 \le x \le 5\) or \(x \ge 8\) (c) same, since \(C(x) \le 10 \iff (x-2)(x-5)(x-8) \le 0\)
1. (a) -8 (b) 1
2. \(P(1) = 0\) ✓
3. (a) \((x - 1)(x - 2)(x - 3)\) (b) \((x + 1)(x - 1)(x + 2)\)
4. \(-\frac{5}{4}\)
5. \(k = 2\)
6. (a) \((x + 3)(x + 2)(x - 1)\) (b) \((x - 2)(2x + 3)(x - 1)\)
7. (a) \(x = 1, 2, 4\) (b) \(x = -1, 2, 3\)
8. \(a = 1, b = -14\), factorised: \((x - 2)(x + 4)(x - 1)\)
9. Proof using Factor Theorem: \(P(2) = 2^n - 2^n = 0\)
10. (a) \((x - 3)(x - 4)(x - 5)\) (b) \(x = 3, 4, 5\) (c) \(x > 5\) or \(3 < x < 4\) (d) height = 3 cm
1. (a) \(x = 5, y = 2\) (b) \(x = 3, y = 2\)
2. (a) \((1,1)\) and \((2,4)\) (b) \((-1,3)\) and \((2,6)\)
3. \((1, \frac{5}{2})\) and \((5, \frac{1}{2})\)
4. \((1,1)\) and \((5,9)\)
5. \(k = 1\)
6. 6 books, 6 pens
7. \(m < -6\) or \(m > 2\)
8. \(m = 0\) (tangent is horizontal), \(c = 1\)
9. (a) 20 of A, 30 of B (b) 30 of A, 20 of B
10. \(c = 5\)
1. (a) 5 (b) 4 (c) -2
2. (a) \(\log_3 35\) (b) \(\log_4 1 = 0\)
3. Graphs are reflections across \(y = x\)
4. (a) \(x = 3\) (b) \(x = 4\) (c) \(x = 2\)
5. 5
6. Approximately 1,314,860 tourists
7. \(x \approx 3.32\)
8. \(x = 4\) (\(x = -2\) is invalid since log undefined for negative)
9. Proof using change of base
10. (a) 700 (b) \(\approx 609\) (c) 200 (d) asymptote \(y = 200\), y-intercept \((0,700)\)
1. (a) \(m = 4, c = -3\) (b) \(m = -\frac{2}{5}, c = 2\) (c) \(m = \frac{3}{4}, c = 2\)
2. (a) \(y = 2x\) (b) \(y = -\frac{2}{3}x + \frac{5}{3}\)
3. (a) \((3,4)\), length = 10 (b) \((5,3)\), length = 10
4. (a) \(y = 2x - 8\) (b) \(y = 3x - 11\)
5. \(y = -x + 10\)
6. Fixed fee = Rs 40, rate = Rs 35 per km
7. \(a = e^2 \approx 7.39, n = 1.5\)
8. \(m = 2, c = -3\)
9. \(m = -\frac{1}{3}, c = -\frac{1}{3}\)
10. (a) \(V = e^9 e^{-0.15t} \approx 8103 e^{-0.15t}\) (b) \(\approx\) Rs 8103 (c) \(\approx\) Rs 3835
1. (a) \(\frac{\pi}{6}\) (b) \(\frac{3\pi}{4}\) (c) \(\frac{5\pi}{4}\)
2. (a) \(30^\circ\) (b) \(120^\circ\) (c) \(225^\circ\)
3. (a) \(2\pi\) cm (b) \(10\pi\) cm²
4. (a) \(8\pi\) cm (b) \(48\pi\) cm² (c) \((8\pi + 24)\) cm
5. (a) 30 cm (b) 360 cm²
6. (a) \(\frac{2\pi}{3}\) (b) \(10\pi\) m (c) \(75\pi\) m²
7. 1 radian
8. \(r = \frac{P \pm \sqrt{P^2 - 16A}}{4}\)
9. \(\theta = \frac{4}{3}\) radians
10. (a) \(\frac{\pi}{2}\) rad (b) \(4\pi\) cm (c) \(8\pi\) cm² (d) \(8\pi\) cm
1. (a) \((x - 2)^2 + (y - 3)^2 = 16\) (b) \((x + 1)^2 + (y - 5)^2 = 36\) (c) \(x^2 + y^2 = 10\)
2. (a) Centre \((3, -2)\), \(r = 4\) (b) Centre \((-4, 1)\), \(r = 5\)
3. Inside (distance squared = 8 < 9)
4. Centre \((3, -2)\), \(r = 5\)
5. Centre \((-5, 3)\), \(r = 7\)
6. \((2 + 2\sqrt{2}, 3 + 4\sqrt{2})\) and \((2 - 2\sqrt{2}, 3 - 4\sqrt{2})\)
7. \(k = -1 \pm \sqrt{17}\)
8. \(y = -\frac{4}{3}x + 10\)
9. (a) Yes (distance from centre to line = 4 < 8) (b) \((15 \pm 4\sqrt{3}, 15.25 \pm 3\sqrt{3})\) (c) \(c = -5\) (tangent condition)
10. \(m < -2\) or \(m > 2\)
1. (a) \(\frac{\sqrt{3}}{2}\) (b) \(\frac{\sqrt{2}}{2}\) (c) \(\frac{1}{\sqrt{3}}\) (d) \(\sqrt{2}\)
2. (a) Amp = 3, Period = \(180^\circ\) (b) Amp = 4, Period = \(720^\circ\)
3. (a) \(60^\circ, 120^\circ\) (b) \(120^\circ, 240^\circ\)
4. Principal axis \(y = -1\), max at \((0,1)\) and \((360^\circ,1)\), min at \((180^\circ,-3)\)
5. \(120^\circ, 300^\circ\)
6. (a) Amp = 200, Principal axis = 500 (b) 700 (c) 12 hours
7. Proof: LHS = \(\frac{2\sin x}{1 - \cos^2 x} = \frac{2\sin x}{\sin^2 x} = \frac{2}{\sin x} = 2\cosec x\)
8. \(x = 30^\circ, 150^\circ, 270^\circ\)
9. Proof: LHS = \(\frac{1 - \sin/\cos}{1 + \sin/\cos} = \frac{\cos - \sin}{\cos + \sin}\)
10. (a) 33 m (b) 3 m (c) 24 minutes (d) \(t = 2\) minutes, 10 minutes
1. (a) 120 (b) 72 (c) 15
2. 120
3. 120
4. 120
5. 1680
6. (a) 495 (b) 210 (c) 245
7. 1680
8. 50,400
9. \({}^{5}P_2 \times {}^{10}P_3 = 20 \times 720 = 14,400\)
10. 1190
1. (a) 26 (b) 185
2. (a) 64 (b) 60
3. \(x^3 + 6x^2 + 12x + 8\)
4. \(a = 6, d = 4\)
5. 765
6. 32
7. 80
8. 805 (approx)
9. (a) \(r = 3\) (b) \(a = 2\) (c) \(S_6 = 728\)
10. 15
11. \(k = 2\)
1. (a) \(3\mathbf{i} + 4\mathbf{j}\) (b) \(-2\mathbf{i} + 5\mathbf{j}\) (c) \(-7\mathbf{j}\)
2. (a) 5 (b) 10 (c) \(\sqrt{2}\)
3. (a) \((7, -2)\) (b) \((-3, -4)\) (c) \((6, -9)\)
4. (a) \((6, 8)\) (b) 10 (c) \((\frac{3}{5}, \frac{4}{5})\)
5. \(k = -\frac{1}{2}\)
6. (a) \((5, 1)\) (b) \(\sqrt{26}\) m/s (c) \(\tan^{-1}(\frac{1}{5}) \approx 11.3^\circ\)
7. \(PQ = QR = 2\sqrt{5}\) (isosceles)
8. \(C = (6.5, 9)\) or \((\frac{13}{2}, 9)\)
9. (a) N \(14.0^\circ\) W (or \(104^\circ\) from east) (b) \(\approx 193.65\) km/h
10. Proof using vector addition of midpoints gives equal and opposite vectors
1. (a) \(12x^3\) (b) \(5x^4 + 6x^2 - 7\) (c) \(2\cos 2x\)
2. 8
3. (a) \(x^4 + C\) (b) \(x^3 + x^2 - 5x + C\) (c) \(-\frac{1}{3}\cos 3x + C\)
4. 9
5. (a) \(e^x(x^2 + 2x)\) (b) \(\frac{1 - x^2}{(x^2 + 1)^2}\) (c) \(\frac{2x}{x^2 + 1}\)
6. \(y = 2x - 6\)
7. \((3, -1)\) is minimum, \((1, 3)\) is maximum
8. \(\frac{dr}{dt} = \frac{1}{8\pi}\) m/min \(\approx 0.0398\) m/min
9. (a) \(\frac{14}{3}\) (b) 1
10. \(f(x) = x^3 + x^2 + 1\)
11. (a) \(v = 3t^2 - 12t + 9\), \(a = 6t - 12\) (b) \(t = 1, 3\) (c) \(s(1) = 6, s(3) = 2\) (d) 6 m
12. \(\frac{4}{3}\) square units
13. Maximum volume = 125 cm³